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Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a simple closed geodesic loop at p . my question is why it should be simple ?

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    $\begingroup$ If by "simple" you mean "has no self-intersections", that statement is false. $\endgroup$
    – Igor Rivin
    Commented Nov 25, 2011 at 12:45
  • $\begingroup$ yeah that's what i mean . what is true or special if you want for negatively curved manifold concerning this subject ( representative of based point class ,length of a representative ,...) a reference would be most welcomed also thank you $\endgroup$
    – student
    Commented Nov 25, 2011 at 12:54

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For a negatively curved manifold, there is a unique geodesic in a free homotopy class, and a unique geodesic broken loop in a homotopy class, and it is the shortest curve. In neither case is the curve necessarily simple. For references, almost any book on differential geometry will work (Cheeger/Ebin, Ballmann/Gromov/Schroeder, Bridson/Haefliger are all good candidates for having a discussion). For curves on surfaces, you might want to check out the little paper of McShane/Rivin in IMRN, which talks about minimal representatives in homology classes...

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